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SIGMA 4 (2008), 046, 9 pages arXiv:0805.4024
https://doi.org/10.3842/SIGMA.2008.046
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics
Hamiltonian Systems Inspired by the Schrödinger Equation
Vasyl Kovalchuk and Jan Jerzy Slawianowski
Institute of Fundamental Technological Research, Polish Academy of Sciences,
21, Swietokrzyska str., 00-049 Warsaw, Poland
Received October 30, 2007, in final form April 25, 2008; Published online May 27, 2008
Abstract
Described is n-level quantum system realized in the n-dimensional ''Hilbert'' space H with the scalar product G taken as a dynamical variable. The most general Lagrangian for the wave function and G is considered. Equations of motion and conservation laws are obtained. Special cases for the free evolution of the wave function with fixed G and the pure dynamics
of G are calculated. The usual, first- and second-order modified Schrödinger equations are obtained.
Key words:
Schrödinger equation; Hamiltonian systems on manifolds of scalar products; n-level quantum systems; scalar product as a dynamical variable; essential non-perturbative nonlinearity; conservation laws; GL(n,C)-invariance.
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